projections and closed subspaces

Theorem 1 - Let X be a Banach spaceMathworldPlanetmath and M a closed subspace. Then,

  • Given a topological complement N of M, there exists a unique continuous projection P onto M such that P(x+y)=x for all xM and yN.

The projection P in the second part of the above theorem is sometimes called the projection onto M along N.

The above result can be further improved for Hilbert spacesMathworldPlanetmath.

Theorem 2 - Let X be a Hilbert space and M a closed subspace. Then, M is topologically complemented in X if and only if there exists an orthogonal projection onto M (which is unique).

Since, by the orthogonal decomposition theorem, a closed subspace of a Hilbert space is always topologically complemented by its orthogonal complementMathworldPlanetmathPlanetmath (X=MM), it follows that

Corollary - Let X be a Hilbert space and M a closed subspace. Then, there exists a unique orthogonal projection onto M. This establishes a bijectiveMathworldPlanetmathPlanetmath correspondence between orthogonal projections and closed subspaces.

Title projections and closed subspaces
Canonical name ProjectionsAndClosedSubspaces
Date of creation 2013-03-22 17:52:57
Last modified on 2013-03-22 17:52:57
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 5
Author asteroid (17536)
Entry type Theorem
Classification msc 46C07
Classification msc 46B20
Synonym projection along a closed subspace
Synonym orthogonal projections onto Hilbert subspaces